Trigonometric functions
Section: Trigonometry | Syllabus: Cambridge IGCSE Mathematics (0580)
Solving Equations of the Form sin x = k and cos x = k
Extended Only
Unlike right-angled triangle trigonometry, which only uses angles between 0° and 90°, equations like sin x = k or cos x = k can have solutions anywhere in the range 0° to 360° - and there are usually two solutions, not one.
- First find the reference angle (the acute angle) using the inverse function on a calculator
- sin x is positive in the 1st and 2nd quadrants (0°-90° and 90°-180°); the two solutions are the reference angle itself, and 180° minus the reference angle
- cos x is positive in the 1st and 4th quadrants (0°-90° and 270°-360°); the two solutions are the reference angle itself, and 360° minus the reference angle
- If k is negative, the same reference-angle method applies, but the solutions come from the quadrants where sin (or cos) is actually negative - sin is negative in the 3rd and 4th quadrants; cos is negative in the 2nd and 3rd quadrants
- Always check both solutions lie within the range given in the question (usually 0° to 360°)
Worked Example: Solving sin x = k
- Question: Solve sin x = 0.6 for 0° ⩽ x ⩽ 360°.
- Step 1: Reference angle: sin^-1(0.6)36.9°
- Step 2: Since sin is positive in the 1st and 2nd quadrants: x=36.9° or x=180-36.9=143.1°
- Answer: x = 36.9° or x = 143.1°
Worked Example: Solving cos x = k
- Question: Solve cos x = 0.35 for 0° ⩽ x ⩽ 360°.
- Step 1: Reference angle: cos^-1(0.35)69.5°
- Step 2: Since cos is positive in the 1st and 4th quadrants: x=69.5° or x=360-69.5=290.5°
- Answer: x = 69.5° or x = 290.5°
Common Mistakes
MistakeOnly giving the reference angle as the final answer, missing the second solution entirely
Fixunless the question restricts the range further, always check for a second solution using the appropriate quadrant rule
MistakeUsing the wrong quadrant rule, e.g. applying the sin rule (180-x) to a cosine equation
Fixsin and cos have different quadrant patterns - sin uses 180° minus the reference angle for its second solution, cos uses 360° minus it
Solving Equations of the Form tan x = k
Extended Only
Equations involving tan x follow a different pattern from sin and cos, since the tangent graph repeats every 180° rather than 360°.
- tan x is positive in the 1st and 3rd quadrants; if k is positive, the two solutions (in the range 0°-360°) are the reference angle, and 180° plus the reference angle
- tan x is negative in the 2nd and 4th quadrants; if k is negative, find the reference angle using the positive version of k, then the two solutions are 180° minus the reference angle, and 360° minus the reference angle
- Because tan repeats every 180° (not 360° like sin and cos), there are still exactly two solutions in a 0°-360° range, but the pattern of quadrants used is different
Worked Example: Solving a Tangent Equation with a Negative Value
- Question: Solve tan x = -1.5 for 0° ⩽ x ⩽ 360°.
- Step 1: Find the reference angle using the positive value: tan^-1(1.5)56.3°
- Step 2: Since tan x is negative, x lies in the 2nd or 4th quadrant: x=180-56.3=123.7° or x=360-56.3=303.7°
- Answer: x = 123.7° or x = 303.7°
Common Mistakes
MistakeApplying the sin/cos quadrant rule to a tangent equation instead of the correct tan pattern
Fixtangent has its own distinct quadrant pattern (1st and 3rd for positive, 2nd and 4th for negative) - learn it separately from the rules for sine and cosine
MistakeTaking the inverse tangent of a negative value directly and using that result as a final answer
Fixalways find the reference angle using the positive magnitude of k first, then apply the correct quadrant rule to find the actual solutions in the given range
Symmetry and Periodicity of Trigonometric Graphs
Extended Only
The shapes of the sine, cosine and tangent graphs explain why trigonometric equations often have more than one solution, and why some equations have infinitely many solutions if the range is not restricted.
- The sine and cosine graphs are periodic with period 360° - their pattern repeats exactly every 360°
- The tangent graph is periodic with period 180° - its pattern repeats twice as often as sine and cosine
- The symmetry of the sine graph gives the identity sin(x) = sin(180-x); the symmetry of the cosine graph gives cos(x) = cos(360-x)
- If a question does not restrict the range of x, a periodic equation has infinitely many solutions, found by adding or subtracting multiples of the period to each solution already found
Worked Example: Using Periodicity to Find an Additional Solution
- Question: One solution to tan x = 2.1 is x = 64.5°. Use the periodicity of the tangent graph to find another solution in the range 0° to 540°.
- Step 1: The tangent graph repeats every 180°
- Step 2: Add 180° to the known solution: 64.5+180=244.5
- Step 3: Add another 180°: 244.5+180=424.5
- Answer: x = 244.5° or x = 424.5° (both within 0°-540°)
Common Mistakes
MistakeAdding 360° (the period of sine and cosine) to a tangent solution, instead of 180°
Fixalways use the correct period for the specific function - 180° for tangent, 360° for sine and cosine
MistakeForgetting to check whether an additional solution found by adding the period actually falls within the range given in the question
Fixafter adding or subtracting the period, always verify the new value is still inside the stated range before including it as a valid answer
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